Chapter 6.8, Problem 37WE

To determine

To find the reciprocal of the given complex number.

Answer to Problem 37WE

  $\frac{1}{2+3i}=\frac{2}{13}-\frac{3}{13}i$

Explanation of Solution

Given information:

Given complex number $2+3i$

We have to simplify $\frac{1}{2+3i}$

We know that $i^2=-1$ and $i=\sqrt{-1}$

Here we shall also use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$

To solve this we have to multiply numerator and denominator by the conjugate of the denominator.

Hence we get,

  $\begin{array}{c}\frac{1}{2+3i}\times \frac{2-3i}{2-3i}=\frac{2-3i}{{\left(2\right)}^2-{\left(3i\right)}^2}\\ =\frac{2-3i}{4-9i^2}\\ =\frac{2-3i}{4+9}\\ =\frac{2-3i}{13}\\ =\frac{2}{13}-\frac{3}{13}i\end{array}$